Two fair dice are rolled, and events A and B are as follows: Event A occurs if the largest number showing is at most 3. Event B occurs if neither 1 nor a 6 is showing Are they Independent?
I said that they are dependent because if Event A occurs where let's say a 1 occurs, then Event B does not occur. However, what happens when 2 occurs because then Event A occurs and Event B occurs as well.
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The definition of $A$ and $B$ being independent is that $P(A\cap B)=P(A)\cdot P(B)$, so you just need to check if that equation is satisfied. We can compute these probabilities as follows.
This table shows all $36$ possible ways to roll two dice. Each cell is labeled with an $A$ if it is part of event $A$, and with a $B$ if it is part of event $B$ (and both letters if it is in the intersection $A\cap B$). $$ \begin{array}{c|c|c|c|c|c|c|} &1&2&3&4&5&6 \\\hline 1&A&A&A \\\hline 2&A&AB&AB&B&B \\\hline 3&A&AB&AB&B&B \\\hline 4&&B&B&B&B \\\hline 5&&B&B&B&B \\\hline 6 \\\hline \end{array} $$ Because each pair of numbers is equally likely, we can determine the probability of each event by counting the number of times it appears in the table, and dividing by the total number of entries, $36$. Therefore,
$$ P(A)=\frac{9}{36},\qquad P(B)=\frac{16}{36},\qquad P(A\cap B)=\frac{4}{36}. $$ You now have enough information to determine whether or not $P(A\cap B)=P(A)\cdot P(B)$, and therefore whether or not $A$ and $B$ are independent.
$A$ and $B$ are independent iff $$\Pr[A \cap B]=\Pr[A] \times \Pr[B],$$ where $\Pr[A \cap B]$ is the probability that both $A$ and $B$ occur simultaneously.
So, given this information, you tell me. Are $A$ and $B$ independent?